A Core Pure mini-paper

1 · 6 marks

Prove by mathematical induction that 1(3 × 1 + 1) + 2(3 × 2 + 1) + ⋯ + n(3n + 1) = n(n + 1)² for every positive integer n.

2 · 6 marks

Let $z=x+iy$.

(a) On an Argand diagram, sketch the locus $|z-(2+i)|=3$, stating its centre and radius. (3 marks)

(b) Find the complex numbers on this locus for which $\operatorname{Re}(z)=2$. Show your working. (3 marks)

3 · 6 marks

The matrix $A=\begin{pmatrix}2&1\\1&3\end{pmatrix}$ represents a linear transformation of the plane, acting on column coordinate vectors. A triangle has vertices $O=(0,0)$, $P=(2,0)$ and $Q=(0,3)$.

(a) Find the image of each vertex and sketch the image triangle. (3 marks)

(b) Find the area of the image triangle. Show how the matrix determines your answer. (3 marks)